Distortion-rate theory for individual sequences

For every individual infinite sequence we define a distortion-rate function which is shown to be an asymptotically attainable lower bound on the distortion that can be achieved for by any finite-state encoder which operates at a fixed output information rate . This is done by means of a coding theorem and its converse. No probabilistic characterization of is assumed. The coding theorem demonstrates the existence of {em universal} encoders which are asymptotically optimal for every infinite sequence over a given finite alphabet. The transmission of individual sequences via a noisy channel with a capacity is also investigated. It is shown that, for every given sequence and any finite-state encoder, the average distortion with respect to the channel statistics is lower bounded by . Furthermore is asymptotically attainable.